A Trajectory Controller for Kite Power Systems
with Wind Gust Handling Capabilities
Manuel C.R.M. Fernandes
1, Gonc¸alo B. Silva
1, Lu´ıs Tiago Paiva
1and Fernando A.C.C. Fontes
11SYSTEC–ISR, Faculdade de Engenharia, Universidade do Porto Rua Dr. Roberto Frias, Porto, Portugal
{up201302946, ee12059, ltpaiva, faf}@fe.up.pt
Keywords: Renewable energy, Airborne Wind Energy, Kite Power Systems, Nonlinear Systems, Path-following control, Safe mode control.
Abstract: In this paper, we address the generation of electrical power using Airborne Wind Energy Systems, comprising a kite connected through a tether to a generator on the ground. We design a controller to steer the kite to follow a pre-defined periodic path, which includes a production mode, a tether retrieval mode, and a safe mode capable of handling wind gusts.
1
INTRODUCTION
Electrical energy is crucial to satisfy modern human needs, while the amount of energy available is directly associated with development. Currently, the majority of electrical energy generated worldwide comes from fossil and nuclear fuels, which are now facing increas-ing societal concerns of environmental sustainability. To address these problems, countries are enact-ing energy policies to encourage the use of renew-able energies. In this context, in the last decades there has been a fast growth and development of re-newable energy systems. Very recently, Portugal’s to-tal renewable energy production in March 2018 ex-ceeded the country’s electricity consumption for the month. Among the renewable energy sources, wind is an important large scale alternative. Wind energy is nowadays mainly extracted on-shore at low heights by wind turbines mounted on towers with a few dozen meters (50-200 m) and, despite the significant num-ber, still growing, of wind farms, most of the existing wind energy remains unexploited since it is available at high altitudes and off-shore.
One of the promising technologies to exploit the stronger and more consistent high altitude and off-shore winds is Airborne Wind Energy (AWE), namely Kite Power Systems (Ahrens et al., 2013; Schmehl, 2018). These systems use a kite, with flexible or rigid wing, that is connected to a generator through a ca-ble (tether). Most of such systems are based on ex-ploiting crosswind kite power described by Loyd in 1980 (Loyd, 1980). The huge power that can
poten-tial be harvested from AWE systems is supported by the facts that wind speeds grow fast with height and that the aerodynamic lift is proportional to the square of the apparent wind velocity
~Flift=1/
2cL(α)Av2a. (1) Thus, the maximum power extraction is obtained when the kite flies at high speeds in direction perpen-dicular to the direction of the wind, which requires this trajectory to be periodic.
In a kite power system with a fixed generator on the ground, which is our case, electrical power is gen-erated as the kite rises by the wind, which causes the cable to be unrolled and subsequently the generator to produce electricity (see Figure 1). Since the cable length is finite, when we reach that limit the cable is retrieved with a minimum energy expenditure and the process is repeated.
To automate this process and guarantee a positive net power output it is necessary to design a controller taking into account variations in wind direction and intensity, among other factors.
Using a 3D simulation model of the kite dynam-ics, considering all the forces acting on it, we design a controller to follow a desired path. Then we ad-dress the problem of handling wind gusts. When wind reaches a certain speed threshold, where tether force might be excessive, we activate gust mode in order to prevent damage to the kite or other components. We do this by stopping the kite reel-out (or reel-in, in case the gust occurs in recovery mode), and directing it up-wards to an equilibrium inclination. We set the angle
Table 1: Nomenclature
A wing reference area of kitem2
RGL rotation matrix from G to L
at tether reel–out accelerationm s−2 RLG rotation matrix from L to G
cD aerodynamic drag coefficient r tether length[m]
cL aerodynamic lift coefficient ρ air densitykg m−3 E energy produced[Ws] T tether tension[N] ~Faer aerodynamic force[N] v
a apparent wind velocitym s−1
~Fdrag drag force[N] v
w wind velocitym s−1
~Fcent centrifugal force[N] v
t tether reel–out velocitym s−1
~Fcor Coriolis force[N] u control vector
~Flift aerodynamic lift force[N] x state vector
~Finert inertial forces[N] α angle of attack[rad]
~Fth tether force[N] φ azimuthal angle[rad]
g gravitational accelerationm s−2 β elevation angle[rad] m mass[kg] ψ roll angle[rad]
P power produced[W] γ reference tracking angle[rad] p kite position [m] τ local tangent plane
Figure 1: Example of a kite power system in the reel-out phase (Argatov et al., 2009)
of attack to a high value, in order to have a low lift force and consequently a low tether tension force.
This technology is still not completely developed at a commercial stage. Therefore, there is a win-dow of opportunity for new challenges, namely in the development of new controllers and continuous improvements to existing methodologies. A similar principle can be applied when the kite moves in an-other fluid, such as water. An example is explored in (Paiva and Fontes, 2017).
In the approach proposed here, the trajectory con-troller acts on the roll angle to change the kite heading direction in order to follow a moving reference point belonging a desired path (see (Prodan et al., 2015), (Caldeira and Fontes, 2010) for other approaches on path-following control). Also, a innovative wind gust mode is developed and tested. The numerical results report that such strategy allows us to handle strong
wind velocities, preventing the system to crash, with-out having to completely retract the kite.
This paper is organized as follows. In section 2, we describe a model for the kite power system. In section 3, we describe the design of a controller that allows the kite to follow a desired path. In section 4, we address the wind gust problem and how we handle it. In section 5, we provide results obtained from our simulations. In section 6, we summarize the conclu-sions.
2
KITE POWER SYSTEM MODEL
Coordinate system
The dynamics of the kite are modelled in a spherical coordinate system positioned at the center of mass of the kite, see (Paiva and Fontes, 2018; Canale et al., 2010). We consider three coordinate systems: Global G: An inertial Cartesian coordinate system
(x, y, z) where the origin is on the ground at the point of attachment of the tether and x is aligned according to the wind direction vw= (vw, 0, 0).
The basis of this coordinate system is(~ex,~ey,~ez).
Local L: A non-inertial spherical coordinate system (r, φ, β). The basis of this coordinate system is (~er,~eφ,~eβ) (Fig. 2).
Body B: A non-inertial Cartesian coordinate system attached to the kite body. The basis of this co-ordinate system is(~e1,~e2,~e3), with ~e1coinciding
x
y
z
r
φ
β
e
φe
βe
rFigure 2: Global and Local coordinate systems (Paiva and Fontes, 2018)
e
1e
2e
3Figure 3: Body coordinate system (Paiva and Fontes, 2018)
~e2in the kite transversal axis pointing to the left
wing tip, and~e3in the kite vertical axis pointing
upwards (Fig. 3). We consider the position
p= x y z = rcos(β) cos(φ) rcos(β) sin(φ) rsin(β) , the rotation matrix from L coordinate system to G RLG=~er ~eφ ~eβ =
=
cos(β) cos(φ) −sin(φ) −sin(β) cos(φ) cos(β) sin(φ) cos(φ) −sin(β) sin(φ)
sin(β) 0 cos(β) , and the rotation matrix from G coordinate system to L
RGL= R−1LG= R > LG.
The apparent wind velocity is va= vw− ˙p. We
assume that its radial component va,r is strictly
pos-itive and that the kite body longitudinal axis aligns naturally with the apparent wind velocity, that is~e1=
−va/kvak. Consider the local tangent plane τ, which
tangent to a sphere centred at the origin and contains the axis~eφand~eβ.
Let ψ be the roll angle measuring rotation around the longitudinal axis (~e1). We consider that initially,
vw ˙p va β r ~Flift ~Fdrag ~Fgrav ~Fth T eβ
Figure 4: Forces acting on the kite (Paiva and Fontes, 2018)
for ψ= 0, ~e2 is in the plane τ. Define ˜e2 to be the
“unrotated” axis, with ˜e2=~e2when ψ= 0. We have
that ˜e2⊥~er, and ˜e2⊥~e1. Thus, we can then define
˜ e2=
~er×~e1
k~er×~e1k
. Finally, we consider the kite body has an anti-clockwise rotation of ψ around the~e1axis: the
roll angle.
We assume here that the roll angle ψ can be con-trolled directly. For example, in a two line kite where d is the distance between attachment points and ∆r is the relative difference between the lengths of each line, we have sin ψ= ∆r/d (cf. (Diehl, 2001)). Using Rodrigues’ formula to rotate ˜e2by ψ around ~e1, we
obtain
~e2= ˜e2cos ψ+ (~e1× ˜e2) sin ψ +~e1(~e1· ˜e2)(1 − cos ψ)
and finally, we define~e3to be~e3=~e1×~e2.
Acting Forces
The total force acting on the kite can be decomposed into the tether, gravity, and aerodynamical compo-nents (see Fig. 4)
m¨p=~Fth+~Fgrav+~Faer(α) (2) where ~Fth= −T ~e r= −T 0 0 L , ~Fgrav= −mg~e z= 0 0 −mg G = −mg sin β 0 −mg cos β L , ~Faer(α) =1/
In the local coordinate system ¨p= ¨r r ¨φcos(β) r ¨β L + −r ˙β2− r ˙φ2cos2(β) 2 ˙r˙φcos(β) − 2r ˙φ ˙βsin(β) 2 ˙r ˙β+ r ˙φ2cos(β)sin(β) L | {z } −1/m~Finert (3) where the second term is −1/m~Finertwith ~Finert repre-senting the inertial forces (centrifugal and Coriolis) in the local coordinate system.
Denoting the total resulting force by F, we have
F= m ¨r r ¨φcos(β) r ¨β
=~Fth+~Fgrav+~Faer(α) +~Finert (4) We assume that the tether acceleration ¨r can be con-trolled directly by at. Denoting by T the tension on
the tether at the base, we have T= Fr− mat. We also
assume that the angle of attack α and the roll an-gle ψ can be controlled directly, possibly by varying the differential lengths of the cable in the kite bridle. Defining the state x=r, φ, β, ˙r, ˙φ, ˙βand the control u= (at, α, ψ), the dynamic equation is
˙x(t) = f (x(t), u(t)) = d dt r φ β ˙r ˙φ ˙ β = ˙r ˙φ ˙ β at 1 mrcos(β)Fφ 1 mrFβ . (5)
2.1
Equilibrium inclination
In strong winds or when the tether is being recoiled sufficiently fast, it is possible to maintain the kite at an equilibrium point with β= β?, φ= 0, and with the angular velocities and angular accelerations equal to zero.
Consider the case of high values of the appar-ent wind speed, when the aerodynamical lift force is much higher than gravity. In such case
~T =~Flift+~Fdrag
and when ˙r= 0, the drag is aligned with the wind, horizontal, and the lift is vertical. Therefore
β?= arctan ~F lift ~Fdrag ! = arctan cL cD
When the angle of attack varies between 0o and 12o degrees, the equilibrium inclination varies be-tween 87oand 85o. For higher angles of attack, we
obtain lower lift to drag ratios and lower equilibrium inclinations. At 13.3owe have a β∗= 76o.
3
TRAJECTORY CONTROLLER
The total cycle consists of a production phase when the tether is reeled-out (˙r> 0) until a maximum value of tether length rmaxis attained. Then, follows a re-traction phase when the tether is reeled-in (˙r< 0) until a minimum value of tether length rminis attained.
In order to have a predictable behaviour and op-timize the energy produced by the system, the kite should track a pre-defined path. In the production phase, we define the path of Figure 5 in the spherical rectangle β ∈[30, 50], φ ∈ [−40, +40] degrees, where the kite should move almost cross wind. In the retrac-tion phase, the kite goes to the vertical plane φ= 0, raises the elevation angle to the equilibrium inclina-tion, and pulls back the tether.
We define the desired production path on a(φ, β) coordinate system, making it independent of the tether length. We use a reference target approach to control the heading direction of the kite. Given the position of the mass-point of the kite p(φ, β) not in the desired path, we determine the closest point to the desired path (A). Then, a reference target B is defined as the point distancing L from A in a forward direction along the path. An auxiliary vector ~L1is computed
between p and B (see Figure 5).
Finally, we compute the angle(γ) between the kite velocity ˙pand ~L1, which serves as a reference to the
desired heading direction adjustment, so that the de-sired path is rejoined at B. In order to control the an-gle(γ) towards zero, we act on the roll angle ψ using a proportional controller: ψ(t) = Kγ(t). The kite trajec-tory control through the heading directions, acting on the roll angle using a proportional controller has been shown to be an adequate steering command (Fagiano et al., 2014).
4
HANDLING WIND GUST
Although the trajectory controller system is resilient enough to support certain strong wind velocities, the resultant tether tension force might be overwhelming for the kiteline or other components. Therefore, for security purposes, a method to handle wind gusts in a robust way is crucial.
L A B p L1 ˙p γ
Figure 5: Trajectory Controller
The chosen response strategy for wind velocities that we consider above the safety limit consists in el-evating the kite towards the zenith, to an equilibrium inclination and centring it (i.e. φ=0), by controlling the roll angle ψ (see Figure 7). Also, the tether reel-ing is stopped and the angle of attack is set to a high value, guaranteeing a low lift force and a high drag, stalling and decelerating the kite. The low lift force minimizes the tether tension and the stalling avoids uncontrollable movements.
When the wind velocity returns to lower values, the system resumes the normal production or retrac-tion phase, depending on which phase was interrupted by the wind gust.
Simulation results are described in the next sec-tion.
5
SIMULATION RESULTS
We consider the simulation parameters of the kite sys-tem defined in table 2.
Table 2: Simulation Parameters
Parameter Value ρ 1.2 kg m−3 vw 10 m s−1 g 9.8 m s−2 m 0.7 kg A 0.28 m2
The aerodynamic coefficients were obtained by making a linear regression from data in (UIUC, 2018).
cL(α) = 0.3 + 0.1α, 0< α < 12
cL(α) = 9.9 − 0.7α, 12 < α < 15
cD(α) = 0.012 + 0.01α
For the simulations we use the software Simulink, where we build the model presented above and design the controllers.
Fixed tether length
We start by simulating how the kite reacts to a wind gust when the tether is on a fixed length. As we can see in Figure 7, the kite follows the prescribed path and when the gust begins, the kite elevates to an equi-librium inclination and is centred in φ. When the wind velocity returns to normal values, the kite descends and resumes to follow the desired path.
As we can see in Figure 8, when the gust mode is activated, the tether force is small, as is desirable when handling strong winds. In this simulation we have first a wind velocity of 10 m s−1, then we in-crease the velocity to 30 m s−1, above the wind gust threshold, and finally we reduce it to 20 m s−1.
Complete production cycle
Now we see an example of a complete production cy-cle, with the reel-out and reel-in of the tether. As we can see on figure 9, as the kite rises and a wind gust occurs, the reeling out of the tether is stopped and the kite is elevated to an equilibrium point, returning to the trajectory as wind velocities return to lower val-ues.
On figure 11 we can see the state variables(r, φ, β). We clearly see the wind gust occurring between [200, 250]s, as r stays constant, φ goes to zero and the elevation angle β rises. On figure 12 we can see the control variables, and as supposed, during the wind gust we have a high angle of attack (α) and ψ is zero since the kite is centered on φ= 0. Finally, on figure 10, we can see energy is produced during the reel-out phase, and a negligible amount is spent during the reel-in phase. During the gust mode, energy is neither produced or spent.
6
CONCLUSIONS
Using a 3D dynamic simulation model of a kite power system, considering the forces acting on it, we de-velop here a method to control the kite to follow a pre-defined path. The prescribed path is defined only by the minimum and maximum values of φ and β, independently of the tether length and reel in veloc-ity. The steering controller in the production path-following phase is based on the adjustment of the roll angle (ψ), working with a constant angle of attack. Although this procedure has shown itself robust for
Figure 6: Simulink System
Figure 7: Kite reacting to wind gust
Figure 8: Tether force
high wind speeds, a safety mode to avoid excessive strain on the tether and other components was devel-oped.
The angle of attack plays an important roll to bal-ance both drag and lift aerodynamic forces and the
Figure 9: Trajectory example with wind gust
resultant tether tension. In the event of wind gusts, we adjust the angle of attack to an appropriate value that decreases the tether tension. Moreover, at the same time the kite is driven towards the zenith where it can handle the excessive energy contained in the wind gust.
ACKNOWLEDGEMENTS
We acknowledge the support of FEDER/- COMPETE2020/NORTE2020/POCI/PIDDAC/-MCTES/FCT funds through grants
SFRH/BPD/-Figure 10: Energy production
Figure 11: State variables -(r, φ, β)
126683/2016, PTDC-EEI-AUT-2933-2014|16858– TOCCATA, and 02/SAICT/2017-31447|POCI-01-0145-FEDER-031447|FCT–UPWIND.
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Figure 12: Control variables -(α, ψ)
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